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The theory of integration which we shall develop in this chapter is re-
stricted to separable metrizable locally compact spaces, this being sufficient
for our purposes in later chapters. We have followed fairly closely the
exposition of N. Bourbaki [22], with the simplifications afforded by our more
restricted hypotheses.

The key results of the theory of integration are Lebesgue's convergence
theorems (13.8), the Fischer-Riesz theorem (13.11.4), the Lebesgue-Nikodym
theorem (13.15.5) and the Lebesgue-Fubini theorem (13.21.7). Unfortunately
it is necessary to include rather a lot of material on upper integrals, measurable
functions and negligible functions, which are indispensable technical tools.
The important properties of certain particular measures on locally com-
pact groups or on differential manifolds will be examined in Chapters XIV
and XVI.

We have also included, amongst the problems, applications of integration
which are not dealt with in the text, especially to ergodic theory and orthogonal
systems. The reader who wishes to go further in these directions should con-
sult [21], [26a], [28], [30], and [31].

Nowadays the purposes of a theory of integration are very different from
what they were at the beginning of this century. If the aim was only to be
able to integrate " very discontinuous " functions, integration would hardly
have gone beyond the rather narrow confines of the " fine " theory of func-
tions of one or more real variables. The reasons for the importance that
Lebesgue's concept of integral has acquired in modern analysis are of quite a
different nature. One is that it leads naturally to the consideration of various
new complete function spaces, which can be conveniently handled precisely
because they are spaces of functions (or of classes of "equivalent" functions)
and not just abstract objects, as is usually the case when one constructs the
completion of a space. Another is that the theorem of Lebesgue-Nikodym

98fr eacn index a.)ecessary and sufficient condition for the equation